I apologize if this question was already asked somwhere else on this website. Let us consider $f:C \to X$ a stable map. This is a point in the moduli space of stable maps. It seems intuitively to me that any holomorphic global section of $f^*(T_X)$ gives a deformation of the stable map. It is well-known, but not clear to me, why the moduli space of stable maps in a neighbourood of this point looks like a closed subvariety in $H^0(C, f^*T_X)$ cut out by $h^1(C, f^*T_X)$ equations. Where do this equations come from? Is there a concrete way to see them? (e.g. some sections of $f^*(T_X)$ give forbidden deformations, or the same deformation,...).

This is a consequence of the deformation theory of morphisms with smooth target: deformations lie in $H^0(f^{\ast}T_X)$ and obstructions lie in $H^1(f^{\ast}T_X)$. This is explicitly covered, for example, in Hartshorne's notes on deformation theory (go to the section on deformations of a morphism).
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Mike SkirvinSep 18 '11 at 14:44

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-1 for the choice of nickname! (just kidding. But I think with such a nickname you'll be perceived as an administrator or something by the new users..)
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QfwfqSep 18 '11 at 15:09

we have a map from deformation space $H^0$ to obstruction space $H^1$ such that its zero locus corresponds to actual deformations. In symplectic geometry, this map is given by Cauchy-Riemann equation.
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Mohammad F. TehraniSep 18 '11 at 15:14

As to why we need to substract $h^1$ from the dimension. Let $(O,m)$ be the local ring of the map $f$ at the moduli space. If the space is smooth at $f$ then the dimension would be $h^0$. But this is not often the case. Generally $h^0$ would be the dimension of Zariski tangent space ( $= \dim_k m/m^2$). If $O$ is not regular, one way to measure its irregularity is by using
infinitesimal lifting property: given a map $\phi: O \to A$, can we lift it to an infinitesimal
extension $A':$ $0\to J \to A' \to A \to 0$. This is the same as extending a deformation of $f$ over $A$ to a higher order.
Such a lift is obstructed by an element $\alpha \in H^1(f^*T_X)\otimes J$ (local to global).

Let $O = (R,m_P)/I$ such that $R$ is regular having same Zariski tangent space. Then
$\dim O$ is roughly $\dim R - \text{rank}(I) = \dim R - \dim I/m_pI$. Now $I/mP_I$ is
the canonical obstruction for $O$ and it can be embedded into $H^1(C,f^*T_X)$, thus the
dimension of $O$ is roughly $\dim R - \dim I/m_pI \approx h^0 -h^1$